Ray shooting and parametric search
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Applications of parametric searching in geometric optimization
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Computing and verifying depth orders
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
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\indent We present efficient algorithms for the {\em ray shooting} problem: Given a collection $\Gamma$ of objects in ${\bf R}^{d}$, build a data structure so that, for a query ray, one can quickly determine the first object of $\Gamma$ hit by the ray. Using the parametric search technique, we reduce this problem to the {\em segment emptiness} problem. For various ray shooting problems, we achieve space/query time tradeoffs of the following type: for some integer $b$ and a parameter $m \, (n \leq m \leq n^{b})$ the queries are answered in time $O(\frac{n}{m^{1/b}}\, \mbox {log} ^{O(1)}n)$, with $O(m^{1+ \epsilon })$ space and preprocessing time $(\varepsilon